Richard Gerd Froese

Professor

Research Classification

Research Interests

Mathematics
Mathematical physics
quantum mechanics
scattering theory
spectral theory

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Master's students

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Graduate Student Supervision

Doctoral Student Supervision (Jan 2008 - May 2021)
Absolutely continuous spectrum for the Anderson Model on trees (2010)

This dissertation is an examination of the absolutely continuous spectrum for the Anderson model on different types of trees. The text is divided into four chapter: an introduction, two main chapters and conclusions. In Chapter 2 the existence of purely absolutely continuous spectrum is proven for the Anderson model on a Cayley tree, or Bethe lattice, of degree K. The method used, a geometric one, is based on some properties of the hyperbolic distance. It is a simplified generalization of a result for K=3 given by R. Froese, D. Hasler and W. Spitzer.In Chapter 3 a similar result is proven for a more general tree which has vertices of degrees 2 and 3 alternating in a periodic manner. The lack of symmetry changes the analysis, making it possible to eliminate one of the steps in the proof for the Cayley tree.

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Master's Student Supervision (2010 - 2020)
Anderson Localization with Self-Avoiding Walk Representation (2012)

The Green’s function contains much information about physical systems. Mathematically, the fractional moment method (FMM) developed by Aizenman and Molchanov connects the Green’s function and the transport of electrons in the Anderson model.Recently, it has been discovered that the Green’s function on a graph can be represented using self-avoiding walks on a graph, which allows us to connect localization properties in the system and graph properties.We discuss FMM in terms of the self-avoiding walks on a general graph with a small number of assumptions.

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